Multiple boundary peak solutions for some singularly perturbed Neumann problems
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Bibliographic record
Abstract
We consider the problem \begin{cases} ɛ^{2}\Delta u - u + f\left(u\right) = 0 & \text{in }\Omega \\ u > 0 \text{ in }\Omega , \quad \partial \text{u}/ \partial v = 0 &\text{on }\partial \Omega , \end{cases} where Ω is a bounded smooth domain in R^N , ɛ > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as ε approaches zero, at a critical point of the mean curvature function H(P) , P \in ∂ Ω . It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P) . In this paper, we prove that for any fixed positive integer K there exist boundary K - peak solutions at a local minimum point of H(P) . This implies that for any smooth and bounded domain there always exist boundary K - peak solutions. We first use the Liapunov–Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes. Résumé Nous considerons le problème \begin{cases} ɛ^{2}\Delta u - u + f\left(u\right) = 0 & \text{in }\Omega \\ u > 0 \text{ in }\Omega , \quad \partial \text{u}/ \partial v = 0 &\text{on }\partial \Omega , \end{cases} où Ω est une domaine bornée avec frontiére lisse en R^N , ɛ > 0 est un parametre petit, et f est surlinéaire et souscritique. Il est bien connu que cette équation possede des solutions avec pointe sur la frontiére telle que la pointe se concentre (quand ε tend vers zero) à une pointe critique de la courbure moyenne H(P) \in ∂ Ω . Il est aussi connu que cette équation possede pleusieurs solutions avec pointes qui se concentrent sur pleusieurs points critiques nondégénerés de H(P) , ou sur pleusieurs maxima locaux de H(P) . Dans ce papier, nous prouvons que, pour chaque entier positif K donné, il existe solutions avec K pointes l̀a frontiére, situées sur un minimum relatif de H(P) . Ceci implique que pour chaque domaine qui est lisse et bornée il existe toujours des solutions avec K pointes à la frontiére. Nous utilisons la methode de Liapunov–Schmidt pour reduire le problème dans une espace de dimension finie. Ensuite, nous utilisons une procédé de maximization pour obtenir les pointes sur la frontiére.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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